HYACINTHOS 24890

 [Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and A'B'C' the pedal triangle of I.

Denote:

Ab, Ac = the orthogonal projections of A on BI, CI, resp.

Mab, Mac = the midpoints of BAb, CAc, resp.

La = the Euler line of IMabMac. Similarly Lb, Lc.

1. La, Lb, Lc are concurrent

2. The parallels to La, Lb, Lc through A, B, C are concurrent.

3. the parallels to La, Lb, Lc through A', B', C' are concurrent.

4. the parallels to La, Lb, Lc through Ia, Ib, Ic ( = excenters) are concurrent.

 

[César Lozada]:

1)      At:

Z1 = 2*a^4-2*(b+c)*a^3-(3*b^2+4*b* c+3*c^2)*a^2+2*(b+c)*(b^2-3*b* c+c^2)*a+(b^2-c^2)^2 : : (barycentrics)

= ((-8*sin(A/2)+2*sin(3*A/2))* cos((B-C)/2)+(cos(A)-1)*cos(B- C)+cos(2*A)-1)*csc(A/2)^2 : :  (trilinears)

= (4*R+r)*X(7)+(4*R+3*r)*X(21)

= midpoint of X(i) and X(j) for these {i,j}: {1,442}, {21,3649}, {2475,10543}

= reflection of X(6675) in X(1125)

= On lines: {1,442}, {7,21}, {10,5719}, {30,551}, {65,6690}, {78,3826}, {79,5426}, {140,5883}, {191,3338}, {497,2475}, {758,942}, {950,3838}, {958,3487}, {962,4428}, {993,6147}, {1376,5703}, {1387,3636}, {1737,6668}, {2098,10587}, {2646,5249}, {3035,3812}, {3601,5880}, {3651,5603}, {3671,4640}, {3771,5835}, {3782,10448}, {3811,9710}, {3897,5434}, {5221,6910}, {5439,6691}, {5441,9614}, {5499,10283}, {5690,10197}, {5787,5886}, {5902,7483}, {6173,7987}

= {X(i),X(j)}-Harmonic conjugate of X(k) for these (i,j,k): (942,1125,4999), (3485,3616,1001)

= [ 2.022889134016318, 1.58502227821853, 1.609700227440945 ]

 

2)      X(79)

3)      X(3649)

4)      X(191)

 

Regards,

César Lozada